graphs and algorithms in communication networks ...€¦ · arie koster, xavier muñoz 1. auflage...

Texts in Theoretical Computer Science. An EATCS Series

Graphs and Algorithms in Communication Networks

Studies in Broadband, Optical, Wireless and Ad Hoc Networks

Bearbeitet vonArie Koster, Xavier Muñoz

1. Auflage 2012. Taschenbuch. xxviii, 426 S. PaperbackISBN 978 3 642 26163 3

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Chapter 2Traffic Grooming: Combinatorial Results andPractical Resolutions

Tibor Cinkler, David Coudert, Michele Flammini, Gianpiero Monaco, LucaMoscardelli, Xavier Munoz, Ignasi Sau, Mordechai Shalom, and Shmuel Zaks

Abstract In an optical network using the wavelength division multiplexing (WDM)technology, routing a request consists of assigning it a route in the physical networkand a wavelength. If each request uses 1/g of the bandwidth of the wavelength, wewill say that the grooming factor is g. That means that on a given edge of the net-work we can groom (group) at most g requests on the same wavelength. With thisconstraint the objective can be either to minimize the number of wavelengths (re-lated to the transmission cost) or minimize the number of Add/Drop Multiplexers(shortly ADM ) used in the network (related to the cost of the nodes).

Here, we first survey the main theoretical results obtained for different grooming

Tibor CinklerDepartment of Telecommunications and Media Informatics, Budapest University of Technologyand Economics, Budapest, Hungary, e-mail: [email protected]

David CoudertMASCOTTE, INRIA, I3S, CNRS UMR6070, University of Nice-Sophia Antipolis, France, e-mail:[email protected]

Ignasi SauMASCOTTE, INRIA, I3S, CNRS UMR6070, University of Nice-Sophia Antipolis, France, andGraph Theory and Combinatorics Group, Department of Applied Mathematics IV, UniversitatPolitecnica de Catalunya, Barcelona, Spain, e-mail: [email protected]

Michele Flammini, Gianpiero Monaco, Luca MoscardelliDipartimento di Informatica, Universita degli Studi dell’Aquila, Via Vetoio, Loc. Coppito, 67100L’Aquila, Italy, e-mail: flammini,gianpiero.monaco,[email protected]

Xavier MunozGraph Theory and Combinatorics Group, Department of Applied Mathematics IV, UniversitatPolitecnica de Catalunya, Barcelona, Spain, e-mail: [email protected]

Mordechai ShalomTel-Hai Academic College, Upper Galilee, 12210, Israel, e-mail: [email protected]

Shmuel ZaksComputer Science Department, Technion, Haifa, Israel, e-mail: [email protected]

A.M.C.A. Koster, X. Munoz (eds.), Graphs and Algorithms in Communication 63Networks, Texts in Theoretical Computer Science. An EATCS Series,DOI 10.1007/978-3-642-02250-0 2, c© Springer-Verlag Berlin Heidelberg 2010

64 T. Cinkler et al.

factors on various topologies: complexity, (in)approximability, optimal construc-tions, approximation algorithms, heuristics, etc. Then, we give an ILP formulationfor multilayer traffic grooming and present some experimental results.

Key words: WDM networks, grooming, ADM, complexity, approximation algo-rithms, heuristics, integer linear programming

2.1 Introduction

Traffic grooming refers to techniques used to organize and simplify routing andswitching in connection-oriented networks, such as WDM (wavelength divisionmultiplexing) or MPLS (Multi-protocol Label Switching) networks, in order to im-prove the usage of the bandwidth and of the components, and therefore to reducethe network cost.

Typically, when establishing a connection in an optical network, one has to installsome equipment at both extremities of the connection, that is, an optical transmitter(laser) at its source and an optical receiver at its destination. But due to the cost ofbuilding, installing, and maintaining devices, it is usually more interesting to use asingle kind of device that can handle both transmission and reception. Such devicesare called Light Termination Equipment, or LTE for short. Thus, every connectionwill involve two distinct LTEs, and two distinct connections may share the sameLTE, provided that one ends at a node while the other starts from that same node.In this context, traffic grooming refers to minimizing the number of LTEs that areneeded in the network to serve all connection requests. The problem of minimizingthe number of LTEs in the network being NP-Hard [58, 84], research effort has con-centrated on the development of efficient approximation algorithms for both staticand online traffic [52, 59, 64, 65, 103]. This is the subject of Section 2.3.

At another level in the network, traffic grooming also refers to techniques usedto combine low-speed traffic streams onto high speed wavelengths in order to mini-mize the network-wide cost in terms of electronic switching. Typically, nodes of thenetwork insert and/or extract the data streams on a wavelength by means of add/dropmultiplexers (ADMs). A WDM or DWDM (dense WDM) optical network can han-dle many wavelengths, each with large bandwidth available. On the other hand, asingle user seldom needs such large bandwidth. Therefore, by using multiplexedaccess such as TDMA (time-division multiple access) or CDMA (code-divisionmultiple access), different users can share the same wavelength, thereby optimiz-ing the bandwidth usage of the network. By using traffic grooming, not only is thebandwidth usage optimized, but also the cost of the network can be cut by reduc-ing the total number of ADMs. Such techniques become increasingly important foremerging network technologies, including SONET/WDM rings and MPLS/MPλSbackbones [108], for which traffic grooming is essential.

In this context, one ADM is needed in a node each time we want to add or droptraffic to or from a wavelength. Therefore, one has to place one ADM in a node for

2 Traffic Grooming: Combinatorial Results and Practical Resolutions 65

each wavelength in which traffic is added or dropped, as can be seen in Figure 2.1.Here, the bandwidth requirement of a traffic stream is expressed as a fraction of thebandwidth offered by a single wavelength, which we call the grooming factor, g, andan ADM is able to drop (or add) up to g unitary traffic streams from (or to) a givenwavelength. Thus, the traffic grooming problem is to minimize the total number ofADMs to be installed in the network in order to accomodate all traffic streams.

Given the general traffic grooming problem of minimizing the total number ofADMs to be installed in the network with respect to the traffic requirement beingNP-complete [21, 101], recent works focus on specific issues. Most of the algo-rithms aim at grooming traffic in such a way that all the traffic between any givenpair of nodes is carried on a minimum number of wavelengths. However, a large partof the network cost depends on the capacity of the multiplexing equipment requiredat each node. Hence, in order to minimize the overall network cost, algorithms haveto take into account a trade-off between the number of wavelengths used and thenumber of required ADMs. Indeed minimizing the number of ADMs is differentfrom minimizing the number of wavelengths: the number of wavelengths and thenumber of ADMs cannot always be simultaneously minimized (see [11, 21, 69] forunitary traffic). Both minimization problems have been considered by many authors.See, for example, the surveys [3, 56] for minimization of the number of wavelengthsand [10, 69, 70, 73, 112, 115] for minimization of ADMs, and [72, 81] for onlineapproaches. Numerical results, heuristics, and tables might be found in [11, 113],and extensions to multicast connection requests in [51, 107].

The reader may also refer to the surveys [27, 57, 89, 117] and books [55, 106,118] for other aspects of traffic grooming that are not considered here; in particular,waveband switching allows switching together a set of predetermined wavelengths(band) issued from one fiber and going to another [18–20, 75, 116]. Various otherconcepts might also been considered as traffic grooming, such as Lighttrails [114],Lighttours [105], or bus labeling [16, 17].

In this chapter, we give an overview of the traffic grooming problems that havebeen addressed within the European project COST 293 GRAAL, and we survey themain exact and approximate results obtained so far for static and online traffic. Wepresent practical approaches for multilayer traffic grooming. The results have beenobtained using a large variety of mathematical tools including graph theory, designtheory, linear programming, combinatorial optimization, and game theory.

This chapter is structured as follows. We start in Section 2.2 with a general defi-nition of the traffic grooming problem, and we give some examples. In Section 2.3we present the modelization and the main results obtained for minimizing the num-ber of LTEs in a network. We continue in Section 2.4 with the more general modelof minimizing the number of ADMs, for which we survey the main combinatorialresults. Then, in Section 2.5, we present an efficient ILP model for multilayer trafficgrooming on general networks subject to general traffic demands. We also presentsome experimental results. We finally conclude this chapter in Section 2.6.


ADM ADM ADM

OADM

Node 1

OADM

Node 2

OADM

Node 3

w 3

w 2

w 1

w 3

w 2

w 1

Fig. 2.1 Placement of ADMs in the network: one ADM for each wavelength used in a node

2.2 Problem Definition and Examples

In this section, we first give precise descriptions and models of LTE and ADM, andthen formalize the traffic grooming problem considered here.

A Light Termination Equipment, LTE, is a device that realizes the interface be-tween the optical domain and the electronic domain. It is constituted of one opticalreceiver and one optical transmitter, so every connection involves two distinct LTEs,one at each endpoint. In this chapter, we assume that the receiver and the transmit-ter of an LTE are tuned on the same wavelength (other assumptions are possible).Also, two distinct connections may share an LTE, provided that one ends at a nodewhile the other starts from that same node, and that both connections are assignedthe same wavelength.

An Add/Drop Multiplexer, ADM, is a device used in synchronous transmissionnetworks (SDHs or SONETs) to add (insert) or drop (remove) lower-data-rate chan-nel traffic from the higher-rate aggregated channel. In optical networks, each ADMcontains an LTE to realize the interface between the optical domain (high-speedchannel) and the electronic domain (lower-speed channels). Thus, an ADM oper-ates on a single high-speed data stream, and so a single wavelength, as can be seenin Figure 2.1. The cost of an ADM is given by its capacity, that is, the maximumnumber of low-speed channels (provided each of them has a unitary bandwidth re-quirement) that can be added or dropped from the wavelength. The capacity of anADM is called the grooming factor or grooming ratio. Finally, note that with groom-ing factor 1, an ADM is nothing other than an LTE.

In optical networks with grooming capabilities, the traffic demands are expressedin terms of low-speed data channels. Thus, one has to assign to each connection re-quest a path and a wavelength with the capacity constraint that at most g (groomingfactor) connection requests are assigned the same wavelength on the same link ofthe network.

An instance of the traffic grooming problem is a triple (G, I,g) where G = (V,E)is a graph modeling the network topology, I is a set of connection requests, and g isa positive integer, namely the grooming factor.


Given a connection request r ∈ I identified by a couple of nodes aiming to com-municate, let Pr be the set of the paths in G connecting the two endpoints relativeto r. We have two main issues:

• the determination of a path system (or path assignment) of (G, I), that is, a func-tion p : I �→ ⋃

r∈I Pr;• the determination of a proper coloring (or wavelength assignment) of (G, I), that

is, a function w : I �→ N+ = {1,2, ...} such that for any edge e ∈ E at most g pathsusing e are colored with the same color.

Some of the results presented in this chapter deal with both issues (Section 2.5),while others, given a path system in the input, focus only on the determination of aproper coloring (Sections 2.3 and 2.4).

Every colored request r ∈ I needs an ADM at each of its endpoint nodes. Follow-ing the above description of ADMs, given a grooming factor g, at most g paths withthe same color, incident to a node through the same edge, can use the same ADM.Furthermore, the same ADM can also be shared by at most g paths with the samecolor, incident to the same node through another incident edge.

The traffic grooming problem is the optimization problem of finding a propercoloring w of (G, I,g) minimizing the total number of ADMs used. Let A(G, I,g) bethe optimal value for such a problem.

To establish ideas we now provide two examples, for uni- and bidirectional rings,respectively.

Unidirectional Ring

Suppose we have a unidirectional ring with four nodes {1,2,3,4} and an all-to-allunitary traffic (one request between each pair of nodes). Since we need one ADMat each extremity of a request, and the routing is unique, we can put requests (i, j)and ( j, i) on the same wavelength, thus using 1/g of the capacity of that wavelengthon the ring. We call such pair of symmetric requests a circle. There are thereforesix circles (i, j) for 1 ≤ i < j ≤ 4. If there is no grooming (i.e., g = 1) we need sixwavelengths (one per circle) and a total of 12 ADMs. If we have a grooming factorg = 2, we can put on the same wavelength two circles, using three or four ADMs ac-cording to whether they share an end node or not. For example, we can put together(1,2) and (2,3) on one wavelength; (1,3) and (3,4) on a second wavelength; and(1,4) and (2,4) on a third wavelength, for a total of nine ADMs, and this is optimal.

Now, if we allow a grooming factor g = 3, we can use only two wavelengths. Ifwe put together on one wavelength (1,2), (2,3), and (3,4) and on the other (1,3),(2,4), and (1,4), we need eight ADMs (solution a, Figure 2(a)); but we can do betterby putting on the first wavelength (1,2), (2,3) and (1,3) and on the second (1,4),(2,4) and (3,4), using seven ADMs (solution b, Figure 2(b)).

More formally, in the above example with N = 4 and g = 3, solution a con-sists of a decomposition of K4 (all circles) into two paths with four vertices each,


4 ADM

1

4 2

3

4 ADM

3

4

1

2

(a) Solution with eight ADMs. Circles (1,2), (2,3), and (3,4) on the firstwavelength, and (1,3), (2,4), and (1,4) on the second wavelength

3 ADM

24

1

3

4 ADM

1

4

3

2

(b) Solution with seven ADMs. Circles (1,2), (2,3), and (1,3) on the firstwavelength, and (1,4), (2,4), and (3,4) on the second wavelength

1

34

1 2

34

2

34

1 234

1 2

3

1 2

(c) Decomposition of K4 associated with the two solutions. Each edge of K4 corre-sponds to a circle

Fig. 2.2 Traffic grooming for a unidirectional ring with four nodes, grooming factor g = 3 all-to-allunitary traffic. Solution 2(a) with eight ADMs, solution 2(b) with seven ADMs, and correspondingdecompositions of K4


[1,2,3,4] and [1,4,2,3], while solution b corresponds to a decomposition into a tri-angle (1,2,3) and a star with edges (1,4), (2,4), and (3,4).

Bidirectional Ring

Consider now a bidirectional ring on five nodes {0,1,2,3,4} with all-to-all unitarytraffic modeled by the complete symmetric digraph K+

5 . In this setting, it is moreinteresting to route requests (i, j) and ( j, i) on different wavelengths with shortestpath routing. For example, with grooming factor g = 3, we can put on a wavelengthrouted clockwise requests (i, i+1 mod 5) and (i, i+2 mod 5), and on a wavelengthrouted counterclockwise requests (i, i−1 mod 5) and (i, i−2 mod 5). We need fiveADMs on each wavelength so overall ten ADMs. But if requests (i, j) and ( j, i)are routed on a same wavelength, then we can put at most three circles (pairs ofsymmetric requests) per wavelength using at least three ADMs. Since K+

5 containsten circles, we need four wavelengths, three of them with three circles and at leastthree ADMs and one of them with at least one circle and two ADMs, so overall 11ADMs.

With grooming factor g = 2, we can put on one wavelength requests (i, i +1 mod 5) and on another wavelength requests (i, i + 2 mod 5). Symmetric requestsare routed similarly in opposite direction and we obtain the partition of Figure 3(b)using overall 20 ADMs. But we can do better by putting on a first wavelength re-quests (i, i + 1 mod 5), (0,2) and (2,4) using five ADMs, and on a second wave-length requests (1,3), (3,5), and (4,1) using four ADMs. We obtain the partition ofFigure 3(c) using overall 2(5+4) = 18 ADMs.

2

1

0 4

3

(a) Set of requets

0

1

0 4

3

2

1

2

3

4

(b) Partition using two times ten ADMs4

1

2

3

40 0

1 3

(c) Partition using two times nine ADMs

Fig. 2.3 Two valid partitions of K+5 when g = 2, using different number of ADMs. Symmetric

requests are routed counterclockwise and partitioned similarly.


2.3 Minimizing the Usage of Light Termination Equipment

In this section, we concentrate on the traffic grooming problem of minimizing thetotal number of LTEs that are needed in the network to serve all connection requests.This problem is NP-hard [58, 84] in general but can be solved in polynomial timefor specific topologies. Also, efficient approximation algorithms have been proposedfor both static and online traffic.

This section is organized as follows. We first consider the path topology wherethe problem can be solved in polynomial time. Then we review efficient approxi-mation algorithms for the ring topology where the problem is already NP-hard, andalso for more general topologies. Finally, we show how game theory can be usefulto solve dynamic and online versions of the problem.

2.3.1 Path

Let the physical topology be the directed path PN with N nodes labelled 1,2, . . . ,N,and N −1 arcs (i, i+1) for 1 ≤ i < N. Let also T TN = {(i, j), 1 ≤ i < j ≤ N} denotea transitive tournament, that is, the set of all requests from left to right.

For any set of requests I ⊆ λT TN , where λ is a positive integer, the problem ofminimizing the number of LTEs on PN can be solved optimally in polynomial timeusing a greedy algorithm. To prove that, it is sufficient to observe that the number ofLTEs needed at node i of PN is equal to max

{d−

I (i),d+I (i)

}, where d−

I (i) (or d+I (i))

denote the indegree (or outdegree) of node i in I, that is, the number of requests {u, i}with u < i (or {i,v} with i < v). We obtain Proposition 2.1, and in Corollary 2.1 wegive the exact number of LTEs when I = T TN .

Proposition 2.1 (Bermond et al. [4]). A(PN , I,1) = ∑N−1i=0 max

{d−

I (i),d+I (i)

}.

Corollary 2.1 (Bermond et al. [4]). A(PN ,T TN ,1) = 3N2−2N−ε4 , where ε = N mod

2.

When the physical topology is a bidirectional path, it is necessary to be preciseabout how LTEs can be used. In particular, one has to be precise about whether itis possible to share a LTE between requests (u, i) and (i,v) with u,v < i, that is, aleft-to-right request ending at i and a right-to-left request starting from i, or not. Ifit is not possible, then the problem can be decomposed into two subproblems on adirected path that will be solved independently. But when such sharing is allowed,the problem has not been addressed in the literature and it is conjectured to be NP-complete.


2.3.2 Ring

The problem of minimizing the number of LTEs in optical networks was introducedin [69] for the unidirectional ring topology. It is proved to be NP-hard independentlyin [58] and [84]. The NP-hardness proofs also apply to bidirectional rings, evenwhen the routing of connection requests is given. An algorithm with approximationratio of 3

2 was presented in [52] for unidirectional and bidirectional rings with givenrouting. This algorithm has a first step (called the preprocessing step) that findscycles in the instance and colors each cycle with a unique color. The remainingrequests are then merged to form chains. This algorithm can also be adapted tothe case where also the routing has to be determined, with the same approximationratio [52].

This technique was improved in [103], showing that if the preprocessing phasetries to remove short cycles first, then an approximation ratio of 10/7 + ε can beachieved. This is improved to 10/7 in [59] using the same technique with a moredetailed analysis.

In [13], we give exact algorithms for the all-to-all set of requests on uni- andbidirectional rings. Surprisingly, these results are obtained using a partition of theset of requests into cycles of lengths 3 and 4.

In [53] and [60] a variant of this problem is considered. In this variant, a pathcan be broken into segments and each segment can be colored using a differentwavelength. Obviously this might incur an additional cost in terms of LTEs, but itallows to reduce significantly the number of wavelength used.

2.3.3 General Topology

In [52] an approximation algorithm was presented for general networks. The algo-rithm has a preprocessing phase where cycles of length at most l are included in thesolution; this algorithm was shown to have performance guarantee of OPT + 1

2 (1+ε)N, 0 < ε ≤ 1

l+2 , where OPT is the cost of an optimal solution and N is the numberof connection requests for any given odd l. A special case of this algorithm is whenthere is no preprocesing (i.e., l = 1). The analysis reduces in this case to OPT + 2

3 N.The dominant part in the running time of the algorithm is the preprocessing phase,which is exponential in l.

In [65] we improve the analysis of the algorithm of [52] and prove a performanceof OPT + 1

2 (1 + ε)N, where 12l+3 ≤ ε ≤ 1

32 (l+2)

. Specifically, we show that the al-

gorithm guarantees an upper bound of OPT + 12 (1 + ε)N for ε ≤ 1

32 (l+2)

, and we

demonstrate a family of instances for which the performance of the algorithm isOPT + 1

2 (1+ ε)N for ε ≥ 12l+3 .

Our analysis sheds more light on the structure and properties of the algorithmby closely examining the structural relation between the solution found by the algo-rithm and an optimal solution for any given instance of the problem. As the running


time of the algorithm is exponential in l, our result implies an improvement in theanalysis of the running time of the algorithm. For any given ε > 0, the exponent ofthe running time needed to guarantee the approximation ratio (3 + ε)/2 is reducedby a factor of 3/2. In addition, in the development of our bounds we address a purelycombinatorial problem, which is of interest by itself.

We also improve the analysis for the special case where there is no preprocesing.In [64] we develop a new technique for the analysis of the upper bound and prove atight bound of OPT + 3

5 N for the performance of this algorithm.

2.3.4 Online Traffic

In many applications the requests arrive at the network online, and we have to assignthem wavelengths so as to minimize the switching cost. In more involved cases wehave also to determine the actual routing for these requests. In these situations, oncean assignment is made the system cannot change it, and the aim is to suggest astrategy that will optimally utilize the network resources. Such a study is thus ofgreat importance in the operation of optical networks.

Formally, an online algorithm is said to be c-competitive if, for any sequence ofinputs, the cost is at most c times that of an optimal off-line algorithm (see [15]).

In [102] we present an online algorithm for the problem of minimizing the num-ber of LTEs, and prove that its competitive ratio is 7

4 . We show that this result is thebest possible in general. Moreover, we show that even for the ring topology networkthere is no online algorithm with competitive ratio better than 7

4 . We show that onthe path topology the competitive ratio of the algorithm is 3

2 . This is the best possi-ble for this topology. The lower bound on the ring topology does not hold when thering is of bounded size. We analyze the triangle topology and show a tight boundof 5

3 for it. The analysis of the upper bounds, as well as those for the lower boundsuse all a variety of proof techniques, which are of interest on their own, and whichmight prove helpful in future research on the topic.

2.3.5 Price of Anarchy

Game Theory and the associated concept of Nash equilibria have recently emergedas a powerful tool for modeling and analyzing a lack of coordination in distributedenvironments. In this setting, each communication request is handled by an agent(or player) selfishly performing moves, i.e., changing her routing strategy in orderto maximize her own benefit. A Nash equilibrium is a solution of the game in whichno agent gains by unilaterally changing her routing strategy. If Nash equilibria arereached in a polynomial number of selfish moves, and finding an improving usermove is a problem solvable in polynomial time, such a non-cooperative processnaturally defines a distributed polynomial-time algorithm. However, due to the lack


of cooperation among the players, Nash equilibria are known not to always optimizethe overall performance. Such a loss has been formalized by the so-called price ofanarchy (or optimistic price of anarchy), defined as the ratio between the cost ofthe worst (or best) Nash equilibrium and the one of a centralized optimal solution.The notion of Nash equilibria goes back to [91]. For about a decade the use of gametheory has gained a lot of attention in numerous computer science directions, in whatis known today as algorithmic game theory (see [92, 100]). The notion of price ofanarchy goes back to [82].

In [54] we consider non-cooperative games in all-optical networks where usersshare the cost of the LTE switches used for realizing given communication patterns.We show that the two fundamental cost sharing methods, Shapley and Egalitarian,induce polynomial converging games with price of anarchy at most 5

3 , regardless ofthe network topology. Such a bound is tight even for rings. Then, we show that ifcollusion of at most k players is allowed, the Egalitarian method yields polynomiallyconverging games with price of collusion between 3

2 and 32 + 1

k . This result is veryinteresting and quite surprising, as the best-known approximation ratio, that is 3

2 +ε ,can be achieved in polynomial time by uncoordinated evolutions of collusion gameswith coalitions of increasing size.

Moreover, with respect to the optimization of the optical spectrum, in [14] weinvestigate the problem in which a provider must determine suitable payment func-tions for non-cooperative agents wishing to communicate so as to induce routingsin Nash equilibria using a low number of wavelengths. We assume three differ-ent information levels specifying the local knowledge that agents may exploit tocompute their payments. Under complete information of all the agents and theirrouting requests, the network provider can compute prices where a Nash equilib-rium is reached such that the assignment is the same as the one computed by acentralized algorithm. If the price to an agent is based only on the wavelengths usedalong connecting paths (minimal level) or along the edges (intermediate level), themost reasonable functions either do not admit equilibria or admit equilibria with theworst possible price of anarchy, that is, the ratio between the number of colors usedby the worst Nash equilibrium and the one used by an optimal solution. However,by suitably restricting the network topology, a constant price of anarchy for chainsand rings and a logarithmic one for trees have been obtained under the minimal andintermediate levels, respectively.

For more information, we refer to Chapter 9.

2.4 Minimizing the Number of Add/Drop Multiplexers

We will now concentrate on the case where the grooming factor is g > 1, forwhich we gave examples in Section 2.2. We first survey the main complexity and(in)approximability results. Then we see that for particular topologies and sets ofconnection requests, it is possible to obtain optimal constructions. We also considerthe case where ADMs are placed a priori.


Let us first clarify the difference between single-hop and multi-hop (or bifurca-tion allowed) routing in this context. With single-hop routing, each request is routedthrough the same wavelength from its source to its destination. This is used for sim-ple network topologies such as directed paths or rings, but not for general topologieswhere multi-hop routing is needed. When multi-hop routing is allowed, a requestmay be switched from one wavelength to another at intermediate nodes. This givesmore flexibility for the traffic aggregation that is useful to optimize simultaneouslythe number of ADMs and wavelengths (see Section 2.5).

2.4.1 Complexity and Inapproximability Results

Determining the NP-completeness of the traffic grooming problem has been an openquestion for many years. It was first proved NP-complete on unidirectional ringsin [21] using a reduction from the Bin Packing problem. Another proof was alsomentioned in [112]. Later, the NP-completeness result has been refined.

More precisely, in [101] the traffic grooming problem is shown to be NP-complete in the strong sense for a given grooming factor g ≥ 2, a network of di-rected path (or unidirectional ring) topology, a set of demands I ⊆ KN , single-hoprouting, and an unbounded number of colors (wavelengths). It is also shown to beNP-complete for rings and for paths for any fixed value g ≥ 2, and when the numberof colors is bounded.

The traffic grooming problem has also been proved NP-complete and hard toapproximate in star networks in [74]. These results have been extended in [62]where a complete characterization of the traffic grooming problem complexity instar networks is given by providing optimal polynomial-time algorithms for g ≤ 2and proving the intractability of the problem for any fixed g > 2.

The first inapproximability result for traffic grooming with fixed values of thegrooming factor g has been obtained in [2], thus answering affirmatively the conjec-ture of [23]. More precisely, it has been proved that traffic grooming on a unidirec-tional ring for fixed g ≥ 1 and traffic grooming on a directed path for fixed g ≥ 2 areAPX-complete. That is, there is no polynomial-time approximation scheme (PTAS)with constant approximation factor for these problems, unless P = NP. Both resultsrely on the fact that finding the maximum number of edge-disjoint triangles in agraph (and more generally cycles of length 2g + 1 in a graph of girth 2g + 1) isAPX-complete.

In particular, this implies that the traffic grooming problem is NP-complete inrings for fixed g ≥ 1 and in paths for fixed g ≥ 2 for an unbounded number ofwavelengths, extending in this way the results of [101].


2.4.2 Approximation Results

The first approximation algorithm for the traffic grooming problem has been de-signed for the ring topology [71]. It is based on a greedy partition of the set ofconnection requests into trees of width at most g and has approximation ratio

√g.

In [63] we present an approximation algorithm for the problem of minimizing thenumber of ADMs on a general network in the case where grooming is allowed. Forevery value of the grooming factor g the running time of the algorithm is polynomialin the input size. The approximation ratio of this algorithm for a wide variety ofnetwork topologies – including the ring topology – is shown to be 2lng+o(lng). In[62] the approximation ratio of the algorithm is shown to be 2ln(δ ·g)+o(ln(δ ·g))for any undirected tree having fixed node degree bound δ , and 2lng + o(lng) forunbounded degree directed trees.

As we have seen above, for general grooming factor g the best approximationalgorithm [63] for the traffic grooming on a ring achieves an approximation factor ofO(logg), but its running time is exponential in g (that is, Ng). However, in practicalapplications such as SONET/SDH WDM rings, which are widely used as backboneoptical networks [57], the grooming factor is equal to 3 or 4, typically when four655 Mbit/s streams are aggregated into one 2.5 Gbit/s wavelength.

It is also important to find good approximation algorithms with running timepolynomial in both N and g. Such approximation algorithm has been proposedin [2], where g is considered as part of the input. To the best of our knowledge,this is the first polynomial-time approximation algorithm for the traffic groomingproblem with an approximation ratio which does not depend on g.

Theorem 2.1 (Amini et al. [2]). There exists a polynomial-time approximation al-gorithm that approximates the traffic grooming problem on a ring within a factor ofO(N1/3 log2 N) for any grooming factor g ≥ 1.

Theorem 2.2 (Amini et al. [2]). There exists a polynomial-time approximation al-gorithm that approximates the traffic grooming problem on a path within a factor ofO(N1/3 log2 N) for any grooming factor g ≥ 2.

Although the performance of this algorithm seems not to be very good at firstsight, in fact it is conjectured in [2] that for the general instance of the problem it isnot possible to get rid of a factor nδ for some constant δ > 0.

Finally, in [2] it is shown that the general scheme of the algorithm yields anO(log2 N)-approximation if the request graph excludes a fixed graph as minor, forexample, if R is planar or of bounded genus. The main theoretical contribution ofthis algorithm is to relate the traffic grooming problem to the dense k-subgraphproblem [61] and the degree constrained subgraph problem [1].


2.4.3 Specific Constructions

For specific grooming factors, sets of requests and topologies, it is possible to giveoptimal constructions (assignment of requests to wavelengths that minimizes thenumber of ADMs). This is typically the case with all-to-all unitary traffic (one uni-tary request between each pair of nodes) where optimal constructions have beenobtained on simple topologies for a specific grooming factor.

In unidirectional rings, all requests are routed clockwise. Therefore, it is possibleto route requests (i, j) and ( j, i) on the same wavelength at the cost of two ADMsand using 1

g of available bandwidth all along the ring. When the set of requestsis symmetric, this is shown to be optimal [11]. Furthermore, in this case, the setof requests can be modeled by an undirected graph, each edge corresponding toa circle, and a subgraph B with g edges corresponds to a valid assignment of gcircles to a wavelength. The number of nodes of B gives the number of ADMs touse on the corresponding wavelength. Therefore, the traffic grooming problem on aundirectional ring with symmetric traffic and grooming factor g can be modelled asthe following partition problem.

Definition 2.1 (Traffic Grooming in Unidirectional Ring with Symmetric Traf-fic).Input: N nodes unidirectional cycle CN , grooming factor g, and set of sym-

metric requests modeled by graph I.Output: Partition of I into subgraphs Bw, 1 ≤ w ≤ W , such that |Bw| ≤ g.Objective: Minimize ∑W

w=1 |V (Bw)|, and the optimum is denoted A(CN , I,g).

This problem is in general NP-complete. However, for the all-to-all unitary set oftraffic requests, I = KN , the complexity of the problem is unknown so far. Indeed,it is clearly a difficult combinatorial problem. Using tools of Design Theory [47],optimal constructions have been obtained for grooming factor g = 3 [5], g = 4 [11,73], g = 5 [7], g = 6 [6], g = 7 [48], and g ≥ N(N − 1)/6 [11]. It has also beensolved for practical values of N and g [9], that is, N ≤ 16 and g = 3,4,12,16,48,64.

When the physical topology is a directed path, the problem has only been solvedfor grooming factor g = 2, with all requests from left to right (transitive tourna-ment, T TN) [4]. As for traffic grooming on a unidirectional ring, the problem can bemodeled as a graph partition problem. The main difficulty here is that the numberof connections in each subgraph is subject to high variation since, for example, allrequests (i, i + 1) may fit in the same subgraph (see [8] for the maximum value forany g ≥ 1), and no suitable tools from graph or design theory have been developedso far. A formal definition of the problem for any valid set of connection requestsis given in Definition 2.2, where load(Bw,e) denotes the number of requests of Bw

routed in the path through edge e.

Definition 2.2 (Traffic Grooming in Directed Path).Input: N nodes directed path PN , grooming factor g, and set of requests I.Output: Partition of I into subgraphs Bw, 1 ≤ w ≤W , such that load(Bw,e) ≤ g

for all e ∈ PN .Objective: Minimize ∑W

w=1 |V (Bw)|, and the optimum is denoted A(PN , I,g).


Table 2.1 Congruence classes of N for some g for which optimal constructions are given

k g N1 1 All values2 3 N ≡ 1,5 mod 123 6 N ≡ 1,7 mod 244 10 N ≡ 1,9 mod 405 15 N ≡ 1,9 mod 306 21 N ≡ 1,13 mod 847 28 N ≡ 1,15 mod 1128 36 N ≡ 1,17 mod 144

Finally, when the physical topology is a bidirectional ring, the routing of therequests has to be taken into account since shortest path routing is not always op-timal in general. However, it has been proved in [12] that symmetric shortest pathrouting allows us to obtain optimal solutions on bidirectional rings with all-to-allunitary traffic. The main results in this case are the following: optimal constructionfor the particular case g = 1 [13]; optimal construction when g = 4,8 [49, 50]; op-timal construction when g = 3 and N ≡ 1,5 mod 12 [12] and when g = k(k + 1)/2for some congruence classes of N summarized in Table 2.1; and construction withapproximation factor 12/11 when g = 2 [12].

2.4.4 A Priori Placement of the Equipment

In this section we study traffic grooming in unidirectional rings considering a widerrange of requests than, for example, a complete graph. The idea is to place theADMs in the nodes with limited knowledge of the graph of requests, for instance,knowledge of only its maximum degree. This model helps the network designerto take into account small traffic variations when deciding where to install ADMs,since in many situations one cannot expect to add or remove equipment at the nodeswhen the requests vary.

Namely, we consider the problem of placing the minimum number of ADMs inthe nodes of a unidirectional ring in such a way that the network could support anyrequest graph with maximum degree bounded by a constant Δ . Note that using thisapproach, as long as the degree of each node does not exceed Δ , the network cansupport a wide range of traffic demands without reconfiguring the equipment placedat the nodes. The problem can be formally stated as follows.

Definition 2.3 (Traffic Grooming in Unidirectional Rings with Bounded-DegreeSymmetric Request Digraph).


Table 2.2 Values of M(g,Δ) found in [90]. The case g = 4 and Δ = 3 is a conjectured value

g \ Δ 1 2 3 4 5 6 . . . Δ

1,2 1 2 3 4 5 6 . . . Δ3 1 2 3 ≥ 3 ≥ 4 ≥ 4 . . . ≥

⌈2Δ3

⌉

4 1 2 2?? ≥ 3 ≥ 4 ≥ 4 . . . ≥⌈

5Δ8

⌉

5 1 2 2 ≥ 3 ≥ 3 ≥ 4 . . . ≥⌈

3Δ5

⌉

g ≥ 5 1 2 2 3 3 4 . . . ≥⌈

g+1gΔ2

⌉

Input: N nodes unidirectional cycle CN , grooming factor g, and a maximumdegree Δ .

Output: An assignment of A(v) ADMs to each node v ∈V (CN), in such a waythat for any request graph I (each edge represents a pair of symmetricrequests) with maximum degree at most Δ , there exists a partition ofI into subgraphs Bλ , 1 ≤ λ ≤Λ , such that:

(i) |E(Bλ )| ≤ g for all λ ; and(ii) each vertex v ∈ V (CN) appears in at most A(v) subgraphs.

Objective: Minimize ∑v∈V (CN) A(v), and the optimum is denoted A(CN ,g,Δ).

This problem has been studied in [90]. It solves the cases corresponding to Δ = 2(for all values of g) and Δ = 3 (except for g = 4), and give upper and lower boundsfor the general case. It also characterizes the function A(CN ,g,Δ), which turns outto be linear in N.

Lemma 2.1 ( [90]). The function A(CN ,g,Δ) is of the form A(CN ,g,Δ) = MN −α ,where M and α are natural numbers depending only on g and Δ .

A summary of the results of [90] is given in Table 2.2, where M(g,Δ) is the smallestinteger such that the inequality A(CN ,g,Δ) ≤ M(g,Δ)N holds for any N ≥ 1.

2.5 Multilayer Traffic Grooming for General Networks

In previous sections we have discussed various aspects of traffic grooming for ringand tree networks. In this section we will discuss the case of more general networktopologies, typically referred to as mesh networks. First we give an overview ofdifferent architectures of practical interest; then we give a survey of different graphmodels used with an ILP formulation and show examples of what can these modelsbe used for.


2.5.1 Multilayer Mesh Networks

If there are multiple network layers “one over the other,” we refer to this structureas “Multilayer” network. It is also referred to as the vertical structure of networks,in contrast to the horizontal, where multiple domains are mutually interconnected.These network layers are not the ISO-OSI layers, where each layer defines somenetwork functionality, but layers that can each provide certain connections or vir-tual connections and that can be established using the same or different networktechnologies.

Examples where the same network technology is used are the old FDM (Fre-quency Division Multiplexed) systems, different ATM (Asynchronous TransferMode) networks with two layers, namely VP and VC layers, and the MPLS (Multi-protocol Label Switching) networks where practically any number of LSPs (LabelSwitched Paths) can be established, where the lower-layer paths are considered aslinks in the upper-layer. In this case, the upper-layer paths share these lower-layerpaths, i.e., they are encapsulated or embedded into these paths.

Examples where different technologies are used are

• PDH over SDH• IP over PoS/MAPOS over SDH over WDM• IP over ATM/MPLS over SDH over WDM• IP over GFP over SDH over OTN over WDM• IP over PPP over Ethernet over ATM-AAL5 over SDH over OTN ...

A multilayer network consists in general of interconnected multilayer and single-layer nodes. The single-layer nodes can be at any network layer, while multilayernodes are those that are attached to two or more layers and/or perform the switchingat two or more layers.

There are two general specifications of such multilayer architectures one referredto as GMPLS (Generalized Multi-protocol Label Switching) by the IETF [86] andthe other ASTN (Automatic Switched Transport Network) by the ITU-T [76].

The IETF GMPLS framework [98] defines the following layers, this time accord-ing to the switching capability, i.e., a layer can be established by different network-ing technologies:

• PSC (Packet Switching Capable, e.g., IP)• L2SC (Layer 2 Switching Capable, e.g., GbEth)• TSC (TDM Switching Capable, e.g., SDH VC-4-4c)• λSC (Wavelength Switching Capable)• WBSC (WaveBand Switching Capable)• FSC (Fiber Switching Capable)

Typically not all these layers are represented in a network, but rather only twoor three of them. Having multiple layers has both advantages and disadvantages.The advantages are that the services can access finer bandwidth granularity andsome additional features of upper-layers only, i.e., for a small ratio of traffic only.The drawbacks are that some functionality is multiplied across layers and that the


complexity of operating multilayer networks is much higher than that of operatingcertain layers separately.

This layered vertical structure is valid for the data plane (DP), i.e., the networkthat carries the user information. However, for configuring and operating such anetwork we need a management and a control plane (MP and CP respectively).

If the DP layers of this vertical structure are run by different operators orproviders, then they must communicate with each other to exchange informationnecessary for routing and other purposes. This vertical communication between MPand CP layers is referred to as Interconnection, and there are three defined Intercon-nection Models: (1) Overlay, (2) Augmented, and (3) Peer model [98].

The Overlay model is a client-server model where the upper (client) layer al-ways adapts to the lower (server) layer. In the case of the Peer model, all necessaryinformation is interchanged between the layers, and they may act together, e.g., inrouting a demand. The Augmented (or hybrid) model is somewhere in between theOverlay and Peer models.

The DP layers in a node can be controlled either each by its own CP instancethat communicates with other layers of that node, or by a single CP instance thatcontrols all the DP layers of that node.

The latter case is feasible only if all the DP layers are run by a single operatoror provider, since there is no need for communication interfaces between the layers.Therefore, a single unified integrated CP can be used for all the layers instead ofthe interconnection, the so-called Integrated Model. The forwarding units of all thelayers of the data plane are connected to a single control plane unit.

Similarly, if such a multilayer network has layers or some parts of certain lay-ers built of interconnected elements of a unique networking technology, or, ratherswitching capability, then the set of these elements is defined by the CCAMP WGof IETF as a Region. A network having multiple different regions is referred to as aMulti-region network [93, 104].

2.5.2 On Grooming in Multilayer Mesh Networks

In switched multilayer transport networks (e.g., ASTN/GMPLS) the traffic demandshave typically bandwidth of orders of magnitude lower than the capacity of wave-length links (λ -links). Therefore, it is not worth assigning exclusive end-to-endwavelength paths (λ -paths) to these demands, i.e., sub-λ granularity is required.Furthermore, the number of wavelengths per fiber is limited and costly. To increasethe throughput of a network with a limited number of wavelengths per fiber, trafficgrooming capability is required in certain nodes.

Here we assume two layers only, i.e., a Wavelength Routing Dense WavelengthDivision Multiplexing (WR-DWDM) Network and one layer built over it. In theWR-DWDM layer, a λ -path connects two physically adjacent or distant nodes.These two physical nodes will seem adjacent for the upper layer built over it. Moregenerally, we can consider this two-layer approach as two layers of a 4–6 layer GM-


PLS/ASTN architecture [98]. However, not only is the framing and layering struc-ture of interest, but the control plane proposed in the GMPLS/ASTN framework isas well.

This upper layer is an “electronic” one, i.e., it can perform multiplexing differ-ent traffic streams into a single λ -path via simultaneous time and space switching.Similarly, it can demultiplex different traffic streams of a single λ -path. Further-more, it can perform re-multiplexing as well: Some of the demultiplexed demandscan be again multiplexed into some other λ -paths and handled together along them.This is often referred to as (traffic) grooming [27]. The electronic layer is requiredfor multiplexing packets coming from different ports (asynchronous time divisionmultiplexing).

This upper electronic layer can be a classical or “next generation” SDH/SONET,MPLS, ATM, GbE, or 10 GbE, or it can be based on any other technology. However,in all cases the network carries mostly IP traffic. The only requirement is that itmust be identical for all traffic streams that have to be demultiplexed, and thenmultiplexed again, since we cannot multiplex, e.g., ATM cells with Ethernet framesdirectly.

2.5.3 Graph Models for Multilayer Grooming

Optical metro and particularly core networks consist of multiple layers, where mul-tiple different networking technologies are stacked one over the other. For simplic-ity, here we assume two layers only, e.g., an IP/MPLS layer over an DWDM layer,both controlled jointly by either one vertically peer-interconnected or one verticallyintegrated GMPLS control plane.

To better utilize network resources, smaller, upper-layer traffic streams are mul-tiplexed (“groomed”) into higher capacity wavelength paths in a distributed waythroughout the network.

In this section we give an overview of known graph models as well as proposesome new graph models that all allow both static and dynamic grooming whileperforming design, dimensioning, configuration, routing, multicasting, traffic engi-neering, and resilience functions.

2.5.3.1 Grooming and Wavelength Assignment for Static Routing

The aim of the Grooming and Wavelength Assignment for Static Routing problem(or, for short, Static Grooming problem) is to find a static configuration of the virtual(logical) topology, and to assign the upper layer demands to this topology. It isassumed that the lower network topology, the number of wavelengths per link, thecapacity of these links, and the traffic matrix is given.


The simplest case of static grooming is when the routing is given, and the routesof certain demands are to be bundled (groomed together) in certain parts of thenetwork and assigned to a wavelength.

In [24, 31, 33], a simple model and various heuristic algorithms based on simu-lated annealing, threshold accepting, and tabu search, as well as a genetic algorithm,are proposed and evaluated. The idea of the model is that each part of a route alongeach link can be assigned to any wavelength if that wavelength has enough freecapacity to accommodate the considered demand. The objective is to have as fewgroomings and wavelength conversions as possible. The elementary heuristic stepis to try out different combinations of assigning a segment of a path to differentwavelength links, where the improvements are accepted with higher probability.

The first model for static grooming where the routing was not given in advancebut performed simultaneously with grooming and wavelength assignment was pro-posed in [32]. Later, a method based on ILP formulation for optimal configurationwas proposed in [43], and due to complexity simple heuristic methods using thesame graph model were proposed in [44].

The wavelength graph model proposed in [32] is as follows. For each fiber linkl = (u,v) with Λ wavelengths from u to v we create Λ arcs, one per wavelength,from vertex ul,λ to vertex vl,λ , 1 ≤ λ ≤ Λ . Thus, node u with Lin incoming linksand Lout outgoing links is associated with vertices ulin,λ and ulout ,λ , 1 ≤ lin ≤ Lin and1 ≤ lout ≤ Lout , and a bipartite digraph from vertices

{ulin,λ

}to vertices

{ulin,λ

}

modeled possible interconnections in network node u. This bipartite digraph will becomplete if it is possible to switch from any wavelength to any other.

The ILP formulation [43] uses the proposed graph model, and finds the minimalcost multi-commodity flow over the graph according to the traffic matrix and thecosts assigned to the edges of the graph. However, the ILP can be solved optimallyfor very small instances only.

Heuristics based on the decomposition into as many shortest path searches asnonzero elements in the traffic matrix were proposed in [44]. Here, empiricalweighting of edges has been also proposed to improve the quality of results. Incontrast to the ILP that gives exact globally optimal results (for very small networkinstances), this approach is an approximation only. It is however easily scalable tovery large networks, since it is based on Dijkstra’s algorithm.

In [110] a heuristic method based on decomposition and iterations has been pro-posed that also contains elements of simulated annealing and tabu search. The ideawas that an element of a traffic matrix is a demand that goes from node a to nodec; however, instead of setting up an end-to-end wavelength path we can use twoshorter lightpaths via an intermediate node b. Then it corresponds to a new trafficmatrix, where elements a to b and b to c are increased by the bandwidth of demanda–c while this a–c entry is decreased by its bandwidth (typically to 0). In this case asimpler graph model was used [22] that originally did not support grooming but onlywavelength routing and assignment in a single-layer network; however, groomingwas handled through the traffic matrix transformations.

The use of Integer Linear Programming ensures that the solution is the globaloptimum in terms of the given objective function. However, as the problem to be


solved becomes more complex, and as the network size increases, ILP can becomeintractable (in particular for NP-hard problems). Still, it is worth using it as a refer-ence, at least for smaller networks. As computing capacity grows, particularly dueto the parallelism of supercomputers, GRIDs, and clusters this will also become aviable solution.

As already mentioned, in [43] an ILP formulation for the wavelength graph hasbeen given. In [25, 26] the formulation has been extended for undirected graphs aswell, with protection either at the upper or at the lower layer.

2.5.3.2 Network Dimensioning and Grooming Node Placement

For a two-layer network, both the layers and the interconnection points between thetwo layers must be dimensioned properly. However, due to the interactions of thelayers, all three must be dimensioned simultaneously, leading to high complexity.

In a network it is not necessary to equip all the nodes with grooming capability.Furthermore, since the O/E (Opto-Electronic) and E/O (Electro-Optical) convertersare very expensive, their numbers should be properly determined to reduce costswhile maintaining proper operation of the network. In [94] three methods are pro-posed for deciding which nodes should perform grooming, and to dimension theirgrooming capacity. The three methods are a greedy approach, a vertex-cover-basedapproach, and a heuristic approach that sorts the nodes according to their eligibilityfor accommodating grooming capability. The three methods have similar perfor-mance. In all cases the wavelength graph has been used.

In [96] a simulation-based iterative heuristic method has been proposed. Its ideais that simulations are run for infinite grooming capability in all nodes, and statistics(probability density functions, or pdfs) of the resource usage are compiled. Basedon these pdfs it is decided in which nodes to keep the grooming capability and howmuch to reduce it. Then simulations are repeated and the whole process continuediteratively.

In [95] the optimization objective was extended to optimise not only the groom-ing capability, but simultaneously the number of wavelengths to be used per fiber aswell.

2.5.3.3 Grooming for Dynamic Routing

“Grooming for dynamic routing” or “dynamic grooming” means, that in an opera-tional network the new demands arrive while the demands already routed get ter-minated sooner or later, i.e., the network changes dynamically. In contrast to staticgrooming this is a less complex problem, since a single demand has to be routed ata time and groomed together with some existing demands; however, it is hard to saywhat is the globally optimal long-term strategy.

Here we discuss some related papers.


In [109] the information multi-domain multilayer (MD-ML) influence of delayof advertisements and inaccuracies due to the topology and link state aggregation isstudied in an MD-ML network. The wavelength graph model has been used; how-ever, this information is available only within the domains. Over domain boundariesa simplified aggregated graph is advertised.

In [38, 39] the advantages and drawbacks are investigated of having both layersswitched according to user demands compared to the case where the WDM systemis fixed, and only rarely reconfigured, while over this virtual topology the demandsare dynamically routed. Here, an enhanced version of the wavelength graph is usedthat we refer to as the Grooming Graph or the Fragment Graph, where a wavelengthpath can be cut into two or more shorter pieces and two or more shorter wavelengthpaths can be concatenated into a longer one to reduce the load of the electronic layer.

Finally, [79] gives an overview of routing demands of different traffic parameters(e.g., very different bandwidths) over multilayer multi-domain networks.

2.5.3.4 VPN, oVPN, VPλN and VON, oVON, VOλN

Virtual Private Networks (VPNs), as well as Virtual Overlay Networks (VONs), arevirtual networks set over real physical networks by separating a part of physicalresources, e.g., link and switching capacities. We will refer to these jointly as VNs(Virtual Networks). When multilayer networks are considered, two main optionscan be differentiated: First, when the virtual topology provided by the lower layer isshared among the VPNs or VONs of the upper layer; second, when the VNs are thevirtual topology, i.e., the wavelength paths are the links of the VNs.

In [85] multi-fiber WDM networks are considered. In this paper full wavelengthconversion capability is assumed in all nodes; therefore, no wavelength continuityconstraint has to be obeyed, but only as many parallel links as the product of thenumber of existing fibers and wavelengths. Heuristics based on decomposition andSuurballe’s shortest pair of paths algorithms (cf. Section 1.5.2.1) are used to deter-mine the best failure-resistant VPNs either demand-by-demand or VPN-by-VPN.

In [28, 87] open VPNs (oVPNs) are optimized by using ILPs while obeyingthe wavelength continuity constraint. ILP formulations for the cases without andwith protection are given. For the case with protection two sub-cases are defined:One with external protection, where the network provider is supposed to protectthe VPN, the other with internal protection, when the VPN is configured in such away that if any link or node fails, the resources of the VPN are used for protection.Finally, in [88] more physical limitations are considered for setting up wavelengthpaths.

2.5.3.5 Grooming for Multicast Traffic

Services like TV or video distribution can be more efficiently provided using point-to-multipoint tree structures rather than many point-to-point connections. These ser-


vices have become increasingly more popular, and the bandwidth used by these ser-vices has also grown, i.e., unlike standard definition digital video, high-definitionvideo is already streamed.

If not a single channel, but rather a bundle of programs is streamed simultane-ously, this bandwidth may achieve or even exhaust the capacity of a single wave-length channel. Therefore, performing the multicast at the optical layer via a splittercan be a much cheaper solution than loading the electronic layer with all the multi-casting.

In [107] multicast trees are obtained by ILP. Breadth and depth constraints areobeyed, and it has been evaluated how many ports and how many wavelengths (re-sources in general) are needed for electronic and optical signal branching and howmany for unicast as a reference.

The wavelength graph model has been used again; however, it had to be modifiedto allow branching of the optical signal, which was not allowed for unicast demands.

In [97] methods for periodical reconfiguration of multicast trees has been pro-posed for two-layer grooming-capable networks. Multicast trees (light trees) changedynamically in time due to the changing of multicast endpoints, which causes degra-dation of the tree. A significant amount of network resources can be saved by regularreconfiguration. The benefit of reconfiguration is investigated for different routingalgorithms and reconfiguration periods.

In [45] various restoration mechanisms for multicast trees are considered.

2.5.3.6 Grooming and Resilience

In two-layer grooming-capable networks the demands can be routed over either theupper or the lower layers, or even using both layers. The same holds for routing theprotection paths of these demands. For dedicated protection only an SRG (SharedRisk Group) disjoint path is to be sought; however, for the case of shared path pro-tection this becomes more complex. Namely, not only the capacity is shared, butalso are the O/E and E/O conversion ports as well as the wavelength paths.

In [25, 26] an ILP formulation for different Dedicated Protection Schemes ispresented, while in [68] a decomposition-based heuristic method has been proposedfor the same purpose. In [66] different methods based on running Dijkstra’s algo-rithm twice or Suurballe’s algorithm for static grooming are presented (cf. Sec-tion 1.5.2.1). In [67] the difference is that dynamic grooming is assumed, i.e., de-mands arrive one by one and are both routed and protected instantly.

In [34] shared protection is proposed and fairness issues in terms of dependenceon bandwidth and distances are investigated.

In [41, 42] a new version of the wavelength graph model has been introduced thatallows not only setting up and tearing down lightpaths, but also fragmenting and de-fragmenting them. The idea is that if there are no free wavelength paths in a node,then an existing wavelength path can be cut (“fragmented”) and the new demand isadded or dropped at that point. If there are two consequent wavelength paths carry-ing the same demand or demands, these can be concatenated, i.e., “defragmented.”


Therefore we refer to this model as “Fragment Graph.” Here, the routing of workingand shared protection paths are considered simultaneously.

In contrast to the previous papers in [78], an Ethernet over WDM overlay is con-sidered, where we compare different configurations of the wavelength path systemof the WDM layer and optimally set up MSTP (Multiple Spanning Tree Protocol)trees of the Ethernet layer.

All the methods discussed in this subsection use the wavelength graph modelexcept the last one, which assumes an overlay model, so a simpler graph is sufficient.

2.5.3.7 Traffic Engineering for Traffic Grooming

The simplest definition of Traffic Engineering (TE) is to “put the traffic whereenough resources are available.” It can be considered as an improved adaptive rout-ing. The adaptivity can be achieved in two ways. First, by setting edge weights inour graph to avoid congestions and higher blocking before they occur (“a priori”).Second, by applying wavelength path fragmentation and defragmentation as alreadyexplained in Subsection 2.5.3.6 to resolve existing congestions for newly arrivingdemands (“a posteriori”). Here we give a short overview of MLTE-(Multilayer Traf-fic Engineering)-related papers.

A general overview of TE in GMPLS controlled multilayer networks is presentedin [111].

Several adaptive edge metrics (weights) for MLTE have been proposed and com-pared in [99], using a simpler graph model than in [80]. Then, adaptive fragmen-tation and defragmentation of wavelength paths is proposed in [35–37] and com-pared to the case with no fragmentation or defragmentation and to the case withOXCs only (i.e., no grooming capability). Next, [29] gives an overview of achieve-ments of Routing TE and resilience in Heterogeneous-GMPLS-controlled networks,while [77] presents experimental results from European testbeds.

Finally, we discuss three papers [30, 40, 83] that perform joint “a priori” and “aposteriori” Traffic Engineering. The idea is that although the fragment graph (FG)is being used for performing “a posteriori” TE, and the edge weights of the FGare as follows. Assuming that roughly no more than half of the demands can beterminated and O/E – E/O converted, the load should be balanced accordingly. i.e.,if there are few demands routed over the network and therefore few wavelengths areused, the longer wavelength paths with less electronic processing (grooming) aremade cheaper. However, if more wavelengths start to be used, but the capacity ofthese wavelengths is not well utilized the cost of grooming will decrease leading toshorter paths over more and shorter fragments.


2.5.3.8 Cross-layer Optimization: Considering Physical Impairments WhileRouting

Often, in networks it is not enough to consider the available resources, but it is alsonecessary to consider the impairments that affect the signal quality at the physicallayer and cause increased Bit Error Rate for services. This is a kind of cross-layeroptimization, where the services are optimized with physical layer constraints.

The first use of grooming to repair the impaired signal was presented in [88]where such VPNs were configured, where the signal quality was satisfactory sincethe physical impairments were considered. In [120] the results were extended forrouting in general. [107, 119] present deeper results on the same topic, while [46]gives an overview of the problem, and proposes an additional method for improvingthe signal quality by increasing the power level of signals that have to go far whiledecreasing the power levels of signals that go to closer destinations in order to avoidthe harmful effect of nonlinear distortions.

2.6 Conclusion

The objective of this chapter was to present an overview of traffic grooming inconnection-oriented networks (mainly in WDM networks) and the wide variety ofmathematical tools used to address this issue. Traffic grooming refers to techniquesused for an efficient sharing of the bandwitdh offers by, e.g., a wavelength, usingTime Division Multiplexing. It is usually associated with the routing of the requestsand the survivability issue in single or multiple failure scenarios. Furthermore, traf-fic requests might be uni- or multicast, the traffic pattern may evolve with time, andthe network could be multilayer. Therefore, traffic grooming is only part of the con-cerns addressed when designing and optimizing a network. But even when restrictedto simple physical topologies (unidirectional path or ring) where the routing is fixedand with small grooming factor, the traffic grooming problem is difficult to solveand to approximate. Also, when all aspects have to be taken into account (trafficgrooming, routing, survivability, and so on), problems to solve are so difficult thatexact solutions are usually no longer expected, and it is essential to develop effi-cient heuristic algorithms. Some of them were presented in Section 2.5. Chapter 3presents the state-of-the-art regarding exact approaches for this problem.

In this research area, several important questions are still open and further re-search are needed. In particular, when optimizing only the number of ADMs inSONET/SDH networks, practical values of the grooming factor are 3 and 4, but thisis reapeated several time from the slower 55 Mbit/s streams to the current 10 Gbit/swavelengths. So, it is important to develop efficient optimization tools for groomingfactors 3 and 4, but also to consider hierarchical problems in which unitary requestsare combined by four into streams that are themselves combined by four, and so on.

In the general context, where traffic grooming is associated with routing and sur-vivability issues, existing heuristic algorithms provide upper bounds without guar-


antee on the quality of the solution. Furthermore, the size of practical problems istoo huge for existing mathematical tools. Therefore, research effort has to be put intothe development of new mathematical tools allowing us to address large instancesand to obtain optimal or near-optimal solutions.

Acknowledgements This work was partially supported by EU COST action 293 – Graphs andAlgorithms in Communication Networks (GRAAL), COST action 291 – Towards Digital OpticalNetworks, and IST FET AEOLUS.

The authors would like to thank specifically European Action COST 293 GRAAL, which of-fered us a fruitful scientific framework to share ideas and expertise with other researchers.

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